Inherent modulation - CiteSeerX

Inherent modulation: a fast chopping method for nulling interferometry a

b

Olivier Absil , Anders Karlsson

and Lisa Kaltenegger

b

a IAGL, University of Liège, B5C, 17 Allée du 6 Août, B-4000 Sart-Tilman, Belgium b ESA-ESTEC, postbus 299, NL-2200AG Noordwijk, The Netherlands

ABSTRACT The reduction of the thermal background emission from the local and exozodiacal dust clouds is a critical element for the success of ESA's space mission, DARWIN. Internal modulation, a technique using fast signal 1 In chopping, isolating the planetary signal from these noise sources, was proposed by Mennesson and Léger. this paper, a short review of internal modulation is given, and new congurations with internal modulation are proposed to reduce the complexity of the beam-combining optics. A modication to the implementation of internal modulation is then investigated. It provides similar performance with a single detector and a greatly simplied optical layout: the number of beam-combiners is reduced by a factor of about two.

The principle

of inherent modulation is dierent from internal modulation in that no sub-interferometers are used: dierent phase shifts are applied to the input beams before recombination such that an asymmetric transmission map is obtained directly, without

±π/2

modulation as used in internal modulation. By combining the phase shifts

and the input beams dierently a transposed transmission map is obtained, allowing the signal to be chopped. During operations, multiplexing between the two interferometers is performed, such that at any time only one interferometer is being used.

Keywords:

Nulling space interferometer, direct planet detection, infrared, thermal background, internal modu-

lation

1. INTRODUCTION Darwin is one of the most challenging space mission ever considered by the European Space Agency (ESA). Its main objectives are to detect Earth-like planets orbiting nearby stars and to characterize their atmospheres by 2 means of low resolution spectroscopy (Léger et al. ). The two diculties associated with direct planet detection are the huge contrast between the planet and the star and their small angular distance. The ux ratio is 9 6 typically of 10 in the visible and of 10 in the mid-infrared, while the angular separation is of 100 mas for 3 a Sun-Earth couple located at 10 parsecs. Infrared nulling interferometry proposed by R. Bracewell in 1978, has been selected by ESA as a baseline for the Darwin mission. This technique achieves both a high angular resolution and a high dynamic range by adjusting the phases of the beams coming from various telescopes to produce a fully destructive interference on the optical axis. Darwin will operate in a wavelength band between 6 and 18 microns where suitable tracers for Earth-type atmospheres can be found. The current mission-model baseline consists of six free-ying 1.5 m telescopes, arranged in a hexagonal conguration about 50 m wide, with in the centre of the array a beam-combining satellite equipped with optical benches. This conguration uses a technique called internal modulation which performs fast signal chopping to isolate the planetary signal from other sources. The main drawback of this conguration is the complexity of the recombination optics: it requires six beam-splitters, twelve beam-combiners and nine achromatic phase shifters. Moreover, six single-pixel detectors are needed to ensure a maximum eciency. In this paper, we propose new congurations to achieve fast signal chopping with a greatly simplied recombination scheme.

Further author information: (Send correspondence to O.A.) O.A.: E-mail: [email protected], Telephone: +32-4-366 97 61 A.K.: E-mail: [email protected], Telephone: +31-71-565 3532 L.K.: E-mail: [email protected]

2. THE NEED FOR MODULATION A nulling interferometer formed of response

R(θ, φ),

n

telescopes with polar coordinates

(Lk , δk ) is characterized by its intensity E(θ, φ), where (θ, φ) are the polar

the square modulus of the complex electrical eld response

coordinates in the plane of the sky with respect to the optical axis (see Fig. 1). Its expression has been derived 4 by Mennesson :
2
n


2 i2π(Lk θ/λ) cos(δk −φ) iφk
R(θ, φ) = |E(θ, φ)| =
Ak e e
, (1)




where

Ak

and

φk




k=1

k . The amplitude coecients are Ai = 1 if the whole collected beam is used in the recombination

are the amplitude modulus and phase shift for telescope

normalized so that for the larger telescope we have

scheme, taking into account the possible losses due to unused beam-combiner outputs. The intensity response projected on the plane of the sky forms the so-called transmission map of the interferometer. This map denes which parts of the eld-of-view are transmitted by constructive interference, and which are blocked by destructive interference. No image is formed: we only measure the integrated ux transmitted by the interference pattern on a single pixel detector. Therefore, there is a priori no distinction between the planetary signal and other sources in the eld-of-view, such as dust, stellar leakage or thermal background. This problem can be solved either by reducing the amount of spurious signals well below the planetary signal, or by extracting the planetary signal by an appropriate technique. P la n e t S ta r f

S k y p la n e

q L T e le s c o p e s p la n e

O

2

d 2

L 1

d 1= 0

Geometrical conguration of the array and the stellar system. Planetary coordinates are given by the o-axis angular direction
θ = (θ, φ). Each telescope of index k is located by means of the vector L
k = (Lk , δk ) in a reference frame with arbitrary origin O, used as a common phase reference. Figure 1.

The rst requirement for a nulling interferometer is thus to achieve a deep central null, in order to extinguish 105 or more. Mennesson and Mariotti5 have shown that, for the target stars and

the stellar light by a factor of

interferometric baselines considered here, this condition can only be achieved if the central transmission of the θ4 (or even better θ6 ), where θ is the o-axis angle when pointing towards the

interferometer is proportional to star.

This requires at least three telescopes in a linear arrangement, or four telescopes in a two-dimensional 6 conguration as proven by Absil. Once the stellar light has been properly cancelled, there are other spurious signals to suppress. These come

from the thermal emission of the zodiacal and exozodiacal dust, and from the interferometer itself. This is the reason why Bracewell proposed to rotate the interferometer around its line-of-sight with a period of about one hour. The planetary ux is then modulated as the planet crosses bright and dark fringes in the transmission map, so that it can be retrieved by synchronous demodulation. However, a low-frequency rotation of the interferometer is not enough to completely cancel these spurious signals, as proven below.

•

Zodiacal Dust Cloud.

The zodiacal cloud is a tenuous disk that surrounds the Sun, composed of dust

released from active comets and colliding asteroids. It is heated to about 300 K at 1 A.U. from the Sun, and is a major contributor to the infrared background seen by Darwin: the total zodiacal ux collected by 1 m-class telescopes is about 1000 times larger than the ux from the exoplanet.

This signal is not

modulated by rotation of the array, so that it can be canceled by signal processing.

However, the real

problem comes from its associated photon noise, and above all from the long-term uctuations of its brightness. These uctuations can be confused with the slowly modulated ux of the planet unless fast signal chopping is used.

•

Exozodiacal Dust Cloud.

Another major source of noise is the exozodiacal cloud surrounding the target

star. Considering a dust cloud similar to ours, its integrated ux over the rst 5 A.U. is about 300 times larger than the planetary emission at 10

µm.

The main problem with the exozodiacal cloud is that its

emission is also modulated by rotation of the array (unless it is seen face-on), so that it can be confused with the exoplanet.

•

Instrumental Background. In order to keep the temperature of the cold optics of Darwin as low as possible,

a planar interferometer concept is required. This avoids a strong thermal coupling between the dierent telescopes. The optical elements will be passively cooled to 40 K, while the baseline detectors require active cooling to 6 K. The thermal emission of the instrument can generate troublesome levels of background signal if its temperature exceed 40 K. Moreover, uctuations of temperature and emissivity in the optical path to the detector will induce uctuations of the thermal background levels, which have to be monitored at a high enough frequency. Therefore, there are two other major requirements for a nulling interferometer: rst, to sample the background uctuations at a high enough frequency, and second, to discriminate between the planet and the exozodiacal cloud. The rst attempts to extract the planetary signal from the exozodiacal emission were made by Angel 7 and Woolf who proposed to use the strong modulation of the exozodiacal emission at twice the interferometer 5 rotation frequency due to its symmetry, and by Mennesson and Mariotti who proposed to break the central symmetry of the interferometer, for example by using ve telescopes regularly located on a circle or on an ellipse. But these congurations did not achieve the required high frequency modulation, and thus were still sensitive α to background uctuations and to other 1/f type noises. This is the reason why internal modulation has been 1 proposed by Mennesson and Léger, following an original idea of J.-M. Mariotti.

3. INTERNAL MODULATION The principle of internal modulation is to divide a telescope array into two (or more) nulling interferometers and to recombine their outputs with a time variable phase shift in order to achieve a fast modulation of the 1 planetary signal (Mennesson and Léger ). The sub-interferometers are formed by dividing the amplitudes of the light beams collected by each telescope (Fig. 2a-b). Both have a real entrance pupil, which means that the phase shifts applied to their light beams are restricted to 0 or 4 (Fig. 2c), but both achieve a θ central transmission.

π.

Their transmission maps have dierent shapes

The outputs of the two sub-interferometers are then recombined on a loss-less beam combiner, which induces a

π/2

phase shift between the input beams in the rst output, and a

−π/2

phase shift in the second output.

The new transmission maps associated to these outputs are both asymmetric, although symmetric to each other

R12 (θ, φ) and R21 (θ, φ) the intensity responses associated to R12 (θ, φ + π) = R21 (θ, φ). This can easily be proven by developing

with respect to their center (Fig. 2d). Denoting

the

two new outputs, this property writes

the

expression of the two intensity responses, which write:

1 |E1 (θ, φ) + eiπ/2 E2 (θ, φ)|2 , 2 1 R21 (θ, φ) = |E1 (θ, φ) + e−iπ/2 E2 (θ, φ)|2 . 2 R12 (θ, φ) =

(2) (3)

By developing the square modulus, we get:

1 1 |E1 (θ, φ)|2 + |E2 (θ, φ)|2 + |E1 (θ, φ)||E2 (θ, φ)| sin(arg E1 (θ, φ) − arg E2 (θ, φ)) , 2 2 1 1 2 R21 (θ, φ) = |E1 (θ, φ)| + |E2 (θ, φ)|2 − |E1 (θ, φ)||E2 (θ, φ)| sin(arg E1 (θ, φ) − arg E2 (θ, φ)) . 2 2

R12 (θ, φ) =

(4) (5)

M o d u la tio n m a p = b e a m - c o m b in e r

T w o s u b - n u llin g in te r fe r o m e te r s , r e a l p u p il (a )

(b )

= a lte r n a to r

T r a n s m is s io n m a p s , w ith q 4 tr a n s m is s io n

T w o a s y m m e tr ic m a p s , c o n ju g a te d b y c e n tra l s y m m e try

(c )

(d )

z 0 p h a s e s h ift z p p h a s e s h ift

(e )

Principle of internal modulation. The outputs of two sub-interferometers are recombined on a lossless beamcombiner to produce the two output signals. Figure 2.

¯ i (θ, φ) for i = 1, 2, Ei (θ, φ + π) = E arg Ei (θ, φ + π) = − arg Ei (θ, φ) and thus R12 (θ, φ + π) =

Now, if the entrance pupil of the two sub-interferometers are real, Eq. 1 gives where the bar denotes complex conjugation, so that

R21 (θ, φ).

Therefore, an extended source with central symmetry such as the exozodiacal cloud has the same

contribution in both outputs. The mean leakage of stellar light through the two transmission maps is also the same. On the other hand, the planet has two dierent contributions because when the planet lies on a bright zone in the rst transmission map, it is located on a dark zone in the second map. By alternately detecting the two outputs on a common detector, we get a nal signal where only the planetary part is modulated. The modulation map (Fig. 2e) shows which places on the sky are strongly modulated (bright zones), and where there is no modulation at all (dark zones). A key advantage of internal modulation is that the switching between the two outputs can be done at a high frequency, so that there is now a clear distinction between the rapid modulation of the planetary signal and the slow uctuations of the background. The two outputs will be continuously monitored by two single-pixel detectors to produce a maximum eciency, and the maximum modulation frequency will be set by the detector read-out frequency. Another advantage of internal modulation is that in principle, the array need not be rotated any more. In practice, a slow rotation of the array will still be used in order to get a better spatial coverage. In order to assess the eciency of nulling congurations with internal modulation, the modulation eciency is dened as the part of the incoming signal which is actually modulated and thus can be retrieved by synchronous demodulation. The expression of modulation eciency reads:

M E(θ, φ) = where

Dk

|R12 (θ, φ) − R21 (θ, φ)| 2 |E1 (θ, φ)| |E2 (θ, φ)| n n = sin(| arg E1 (θ, φ) − arg E2 (θ, φ)|) , 2 2 D k k=1 k=1 Dk

is the normalized diameter of telescope

k

(6)

(normalization to 1 for the larger telescope). This denition

of modulation eciency allows an easy comparison between dierent congurations because it is independent of the number and size of telescopes, since the amplitude responses

Ei (θ, φ)

are normalized to the size of the larger

telescope of the array.

1 Some congurations with internal modulation have already been proposed by Mennesson and Léger and 8 by Karlsson and Mennesson. These include the so-called Robin Laurance congurations, which satisfy the following conditions:

•

All telescopes are at equal distance from the beam combiner spacecraft to equalize the optical paths,

•

All telescopes are in one plane (perpendicular to the line-of-sight) to reduce thermal coupling,

•

The central transmission is proportional to

θ4

(or

θ6 )

to cancel the star light eciently,

•

The only applied phase shift to each sub-interferometer is

π

•

All telescopes have the same diameter to reduce manufacturing costs.

(real entrance pupils),

Robin Laurance's congurations are referred to as RLx(m1 ,m2 ,. . .,mn ), where meters,

n

the number of telescopes and

mi

x is the number of sub-interfero-

the relative telescope size, starting with the largest one. The current

baseline for the Darwin mission is the RL3(3,2,0,1,0,2) conguration with six telescopes arranged in a regular 8 hexagonal conguration, as proposed by Karlsson and Mennesson. It achieves internal modulation between three Generalized Angel Crosses (GAC). It has a maximum modulation eciency of 0.29, which means that in the best case, where the planet is located on a modulation maximum, only 29% of the planetary signal is actually modulated and can be retrieved by signal processing. The mean modulation eciency on a eld-of-view of

300 × 300

mas is of 12% for an array 25 meters in radius and a working wavelength of 10

µm.

This is the

relevant gure during the detection phase where the planet position is not known and the whole eld-of-view has to be investigated. Note that a minimum separation of 20 m is required between the free-ying spacecrafts to reduce thermal coupling and to avoid collisions. Larger separations are preferred. The main problem associated to this conguration, in addition to its rather low modulation eciency, is the complexity of the optical recombination scheme to be implemented: it needs not less than nine beam-splitters with ratios of 5:4, 4:1 and 1:1, as well as nine beam-combiners and associated achromatic phase shifters to 8 produce the three GAC outputs (see Karlsson and Mennesson ). Three more beam-combiners are needed to produce the six nal outputs.

Moreover, six detectors are needed for an optimum eciency.

Therefore, new

congurations have to be found, yielding a higher modulation eciency with a simpler recombination scheme. This would improve the reliability and decrease the cost of the mission.

4. NEW CONFIGURATIONS WITH INTERNAL MODULATION Two main types of congurations have been considered so far: the circular ones by Mennesson and collaborators, 7 and the linear ones mainly by Angel and Woolf. Both types of congurations can benet from internal 4 modulation. In order to achieve both a θ central transmission and internal modulation, linear (resp. circular) 6 congurations need at least four (resp. ve) telescopes, as proven by Absil. Circular congurations are very convenient for space interferometry:

if the beam-combiner spacecraft is

located at the center of the circle and the plane of the array perpendicular to the line-of-sight, the optical paths traveled by each light beam are equal, so that long delay lines are not required. A diculty associated with linear congurations is that the light beams have to be reected on neighboring telescopes in order to equalize the external optical delays (see Fig. 3). This could exaggeratedly constrain the control laws on their positions and attitudes.

Figure 3.

Optical path equalization in the cases of circular and linear congurations.

In the following paragraphs, we present a new circular conguration combining a high modulation eciency and a rather simple recombination scheme. It is then compared with a very interesting linear conguration.

4.1. Circular congurations 6 Circular congurations with ve and six telescopes have been comprehensively studied by Absil.

The best of

them seems to be the so-called bow-tie, also referred to as the Liégeoise conguration, with six telescopes in an irregular hexagonal shape (Fig. 4). This Robin Laurance congurationRL2(1,1,0.707,0,0,0.707)performs internal modulation between two identical sub-interferometers (GACs).

Its maximum modulation eciency

reaches 65%, that is, more than twice that of the original RL3(3,2,0,1,0,2) hexagon. eciency is still rather low (13%) on a

Its mean modulation

300×300 mas eld-of-view, but we can manage to get a high eciency across

the whole habitable zone of the target star by tuning the interferometer baseline (Fig. 5). The recombination scheme of the bow-tie (Fig. 4) is much easier to implement than the RL3(3,2,0,1,0,2) conguration, in that it involves only two 1:1 beam-splitters, six beam-combiners with associated achromatic phase shifters, and an additional beam-combiner to form the two nal outputs.

Two detectors are needed to ensure a maximum

eciency. T 3

T 1

T 2

T 2

T 3

T 4

T 6

T 5

D

= b e C D ( C

4 5 °

T 4

1

T 1

1 1

1

a m = c o = d e w ith

c o m n s tr s tru a p

b in e u c iv e c tiv e p h a s

r + o u o u e s

p h a s e - s h ifte r tp u t tp u t h ift)

= b e a m - c o m b in e r

T 5

T 6 D D

1

-1 - 1 /- 2

- 1 /- 2

1 /- 2

= |E 1|2

R

G A C 1 C

D

1

C

R

1 2

= 0 .5 * |E 1+ i E 2|2

R

2 1

= 0 .5 * |E 1- i E 2|2

C

1 /- 2

+ D

1

D

C D

-1

R

G A C 2

2

= |E 2|2

C

C

Left: The bow-tie conguration with its two sub-interferometers (Generalized Angel Crosses) and the relative signed amplitudes for each beam. Right: Recombination scheme for the bow-tie interferometer. The nal outputs are characterized by their transmission maps R12 and R21 . Note that one half of the intensity is lost at each beam-combiner because only one output is used. Figure 4.

The modulation map of this new conguration is presented in Fig. 5. maxima, each with a peak modulation eciency of 65%. wavelength of 10

µm,

It shows eight bright modulation

With an array radius of 25 m and an observation

these maxima are located at 50 mas and 80 mas of the optical axis. This is also the typical

angular separation between an Earth-like planet and its host star located at 10 pc. The right-hand side of Fig. 5 shows the result of a continuous rotation of the interferometer on the modulation map: we get two modulation crowns, with a peak eciency of 65% for each. The coverage of the habitable zone is quite good, provided that the array radius is tuned to the desired target. A slow continuous rotation of the telescope array is not expected to increase the complexity of the mission, nor to require air important quantity of fuel. With the bow-tie conguration, the time required to detect and characterize exoplanets by spectroscopy is reduced by a factor of four as compared to the RL3(3,2,1,0,1,0) conguration. This should allow to observe up to 500 main sequence stars during the mission lifetime instead of the 150 stars initially planned. This advantage adds to the relative simplicity of the recombination scheme, making the bow-tie a very good candidate for the nal Darwin architecture.

4.2. Linear congurations 9 Linear congurations achieving internal modulation have been reviewed by Lawson et al, and have been 6 further investigated by Absil. The most promising of them is presented in Fig. 6. This conguration is dubbed LinEqual because of its four equal-size telescopes.

It achieves internal modulation between two irregular

Degenerate Angel Crosses (DAC), which shapes have been chosen so that all telescopes have the same diameter. The recombination scheme of LinEqual is also presented in Fig. 6.

It uses only four beam-combiners and

associated achromatic phase shifters instead of six for the bow-tie. Note that the beam coming from the centre telescope of each DAC undergoes only one beam-combination, so that it is less attenuated than others.

Left: Modulation map of the bow-tie interferometer over a 300 × 300 mas eld-of-view for an array radius of 25 m and an observation wavelength of 10 µm (isocontours at 20, 50 and 80% of maximum). Right: Modulation map obtained by continuous rotation of the interferometer around its line-of-sight.

Figure 5.

T 1

T 3

T 2

T 4

T 1

T 2

T 4

T 3

D

= b e C D ( C

3 L L

3

1

L

8 8

1

a m = c o = d e w ith

c o m n s tr s tru a p

b in e u c iv e c tiv e p h a s

r + o u o u e s

p h a s e - s h ifte r tp u t tp u t h ift)

= b e a m - c o m b in e r

-2 - 2 1

D

+

= |E 1|2

R

D A C 1 D

1

R

1 2

= 0 .5 * |E 1+ i E 2|2

R

2 1

= 0 .5 * |E 1- i E 2|2

C C

1

-2 - 2

D

3

R

D A C 2 D C

2

= |E 2|2

C

Figure 6. Left: The LinEqual conguration with its two sub-interferometers (Degenerated Angel Crosses) and the relative signed amplitudes for each beam. Right: Recombination scheme for the LinEqual interferometer. The nal outputs are characterized by their transmission maps R12 and R21 .

The modulation map of LinEqual is displayed in Fig. 7.

The line-shaped map is characteristic of linear

congurations. The maximum modulation eciency of LinEqual is of 67%, almost the same as for the bow-tie. Its mean modulation eciency across the

300 × 300

mas eld-of-view reaches 19%, making LinEqual somewhat

more ecient than the bow-tie during the detection phase, where the planet position is unknown and the whole

eld-of-view has to be investigated. A continuous rotation of the interferometer around its line of-sight is used to get a uniform sky coverage, as shown on the right-hand side of Fig. 7. In order to keep the separation between free-ying spacecrafts larger than 20 meters, an array length of 100 meters is needed. wavelength of 10

Assuming a working

µm as usual, the maximum of modulation eciency is reached at about 60 mas from the optical

axis, which makes this conguration appropriate for habitable planet detection around Sun-like stars at 10 pc.

Left: Modulation map of the LinEqual interferometer over a 300 × 300 mas eld-of-view for an array length of 100 m and an observation wavelength of 10 µm (isocontours at 20, 50 and 80% of maximum). Right: Modulation map obtained by continuous rotation of the interferometer around its line-of-sight. Figure 7.

5. INHERENT MODULATION We have seen how internal modulation uses two asymmetric transmission maps conjugated to one another to achieve a modulation of the planetary signal without modulating other signals. Keeping this basic principle, a new way to produce the conjugated maps has been found. The principle of inherent modulation is to combine a number of phase shifted beams, using two opposite sets of phase shifts to produce the two output signals (Fig. 8). The beam-combiner which was used to produce the two conjugated maps is not required any more. With inherent modulation, the two outputs are not formed simultaneously, but one after another: all the light is alternately sent to a set of phase shifting devices or to the other (Fig. 9). Therefore only one detector is needed, which reduces the detector noise by a factor of

√ 2.

It is noteworthy that some of the achromatic phase shifts are dierent from 0 or not fractions of

π.

π , and that they are generally

The entrance pupil of the interferometer is thus complex, which means that the amplitudes iφ of the telescopes are multiplied by a set of complex coecients e k before recombination. A complex entrance

4 p /5

-2 p /5

0 T 2

T 3

2 p /5

-4 p /5

2 p /5

-4 p /5

T 1

T 4

T 5

0 M o d u la tio n m a p -2 p /5

4 p /5

T w o s u b - n u llin g in te r fe r o m e te r s , c o m p le x c o n ju g a te d p u p il

Figure 8.

= fo ld in g m ir r o r

T w o a s y m m e tr ic m a p s , c o n ju g a te d b y c e n tra l s y m m e try

= a lte n a to r

Principle of inherent modulation. Only one interferometer is used at a time.

6 pupil is a necessaryand almost sucientcondition to get an asymmetric transmission map (Absil ). As in the case of internal modulation the two transmission maps are symmetric to each other with respect to their center, as proven by means of Eq. 1:

R1 (θ, φ + π)


2
n



Ak e−i2π(Lk θ/λ) cos(δk −φ) eiφk



k=1
2
n


i2π(Lk θ/λ) cos(δk −φ) −iφk
Ak e e





= =

(7)

(8)

k=1

¯ 2 (θ, φ)|2 = R2 (θ, φ) |E

=

(9)

A major advantage of inherent modulation is the relative simplicity of its recombination scheme.

This is

illustrated in Fig. 9, where the recombination schemes for a four-telescope linear array using internal and inherent modulations are compared. With inherent modulation, the beam-splitters are replaced by folding mirrors. Eight phase-shifting devices and three beam-combiners are used instead of six beam-combiners and associated phase shifters followed by another beam-combiner. The number of phase-shifting devices can sometimes be reduced if some of the

φi

are equal to some of the

−φj ,

which is the case in the example of Fig. 8. Another example is the

RL3(3,2,1,0,1,0) conguration, which can be implemented on ve beam-combiners instead of twelve.

T 1

T 2

T 3

T 4

f D

D

R C

D

T 1

C = c o n s tr u c iv e o u tp u t D = d e s tr u c tiv e o u tp u t ( w ith a p p h a s e s h ift)

1

1

T 2

-f 1

f 2

T 3

-f 2

f 3

T 4

-f 3

f 4

-f 4

= |E 1|2

C

R

1 2

= 0 .5 * |E 1+ i E 2|2

R

2 1

= 0 .5 * |E 1- i E 2|2

C D

D C

R D C

2

= |E 2|2

C

Recombination schemes for internal (left) and inherent (right) modulations in the case of a tour-telescope linen array. The number of beam-combinations is reduced from seven to there by using inherent modulation. Figure 9.

It can be demonstrated that all transmission maps obtained by published nulling techniques can be obtained without actually using sub-interferometers and modulation between sub-interferometers. In order to derive the sets of phase shifts and amplitudes for inherent modulation, starting from a known conguration with internal modulation, one calculates the contribution, in terms of amplitude and phase, from each input telescope to the

output signal on the detector. The transposed transmission map is obtained by considering rather than

+π/2

−π/2

modulation

between the two sub-interferometers. All current congurations using inherent modulation

are derived from known congurations with internal modulation. However, there is in principle no reason why a conguration using inherent modulation would necessarily have a counterpart based on modulation between sub-interferometers. Eorts are currently undertaken in order to identify such congurations.

5.1. Circular congurations 4 Circular arrays with a complex entrance pupil can achieve the necessary θ central transmission only if ve 6 telescopes or more are used (see Absil ). Recombining ve telescopes at the same time forces us to use a 5 third order recombination scheme (Mennesson and Mariotti ). This type of beam-combining scheme has a low eciency:

the recombination eciency is of

5/8,

while it is equal to 1 for a four-telescope array.

The

recombination eciency is the maximum constructive interference one can expect from a given recombination scheme. Therefore, the modulation eciency of circular arrays with inherent modulation is generally low. The simpler circular array with inherent modulation is the Nils interferometer shown in Fig. 8. Its modulation map has a very interesting shape allowing a good sky coverage without any array rotation. However, the maximum modulation eciency is only 30%, taking into account the fact that both outputs are only measured half of the time. Circular congurations with inherent modulation are not expected be signicantly more ecient than this one, but their rather simple recombination schemes still make them good candidates for Darwin. T 1

4

T 2

1

T 5

3 p /1 0 E

E a

b

T 3

p /5

T 4

3 p /5 E

E c

d

A

= ( E b + iE c ) /s q r t( 2 ) A

E B

= ( E d + iE e ) /s q r t( 2 )

E C

= (E

E

O U T

A

= (E

+ iE C

B

)/s q rt(2 )

+ iE a ) /s q r t( 2 )

9 p /1 0 E e

E E

E

B

E C

E

O U T

Third-order recombination scheme for the Nils interferometer. The ipping mirrors are used to send alternately the light to dierent phase-shifting devices. One half of the intensity is lost at each beam-combiner because only one output is used. The expressions of the electrical eld after each beam-combiner are given in the top-right corner. Figure 10.

The recombination scheme of Nils is presented in Fig. 10. The achromatic phase-shifting devices are chosen so that the appropriate phase shifts are obtained after recombination, as proven by the expression of the output electrical eld, including the

EOUT

= =

π/2

phase shift induced by the beam-combiners:

   1  √ E1 + eiπ/2 ei3π/10 E2 + ei(π/5+π/2) E5 + eiπ/2 ei3π/5 E3 + ei(9π/10+π/2) E4 2 2  1  √ E1 + ei4π/5 E2 + e−i4π/5 E5 + e−i2π/5 E3 + ei2π/5 E4 2 2

(10)

(11)

Another interesting circular conguration is the inherent bow-tie, with six equal-size telescopes positioned at

−π/4, 0, π/4, 3π/4, π

and

5π/4 with

phase shifts of

3π/4, −2π/4, π/4, −3π/4, 2π/4 and −π/4.

The transposed

transmission map is obtained by applying the opposite phase shifts. This conguration has the same modulation map as the original bow-tie (Fig. 5), but the maximum eciency is only of 33%, i.e. half of the original one. The phase shifts could be implemented using for example a stack of phase-plates or by a combination of achromatic

π -phase shifter and geometric delays.

In the latter case, the positive phase shifts are implemented by a geometric

delay of length corresponding to the required phase shift on a central wavelength (e.g. middle wavelength). The negative phase shifts are implemented in a similar way after having phase shifted the beams by

π

using an

achromatic phase shifter. The method builds on the principle that the chromatic error introduced by the delay lines belonging to the positive phase shifts are compensated by the errors of the delays lines of the negative phase 5 shifts. This method does not achieve the required 10 rejection rate across the whole 6-20 µm band, but could be implemented by dividing the domain into three smaller wavebands.

5.2. Linear arrays 4 In order to achieve a θ transmission with a complex-pupil linear array, four telescopes are needed at least 6 (see Absil ). The second order recombination scheme used for four-telescopes arrays make them very attractive because the amount of light lost during the recombination process is much lower than for ve-telescope arrays. Their eciency is thus expected to be higher than for circular arrays.

One of the best linear congurations

is presented in table 1. It is composed of two big and two small telescopes (diameter ratio of 0.65), each one at equal distance of its neighbors. The maximum modulation eciency of the LinOpt conguration is of 46%

Table 1.

Parameters for the LinOpt conguration. T1

T2

T3

T4

Relative diameter

D1 = 0.65

D2 = 1

D3 = 1

D4 = 1

Position

L1 = 0

Phase shift

and the mean eciency of 17% on a

m

φ1 = −0.511 300 × 300

L2 = 20

m

φ2 = 3.142

L3 = 40

m

φ3 = 1.267

L4 = 60

m

φ4 = 4.920

mas eld-of-view. Its modulation map is presented in Fig. 11,

showing two maxima of modulation located at 40 and 70 mas of the centre for an array length of 60 m. The recombination scheme of LinOpt is presented on the right-hand side of Fig. 9.

Eight phase-shifting devices

and three beam-combiners are needed to produce the nal output. A drawback of this conguration is the two dierent sizes for the telescopes, which could increase the manufacturing costs.

6. CONCLUSION In this paper, new congurations with internal modulation have been investigated, leading to a simpler optical layout and increased performances: the maximum modulation eciency reaches about 65%, i.e. two times larger than for previous congurations. Inherent modulation, a new way to implement internal modulation, has been proposed, giving an even more simple layout provided that phase shifts dierent from

π

can be achieved. This

type of modulation is particularly adapted to linear congurations with four telescopes, yielding a maximum modulation eciency of almost 50%. These new congurations decrease the complexity of the mission by a factor of 2, and increase the number of potential targets during the mission lifetime by a factor of 4.

ACKNOWLEDGMENTS This Research was supported through a European Community Marie Curie Fellowship.

The author is solely

responsible for the information communicated, published or disseminated. It does not represent the opinion of the Community. The Community is not responsible for any use that might be made of data appearing therein.

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123, p. 249,

274, p. 780, 1978.

Figure 11. Left: Modulation map of the LinOpt interferometer over a 300 × 300 mas eld-of-view for an array length of 60 m and an observation wavelength of 10 µm (isocontours at 20, 50 and 80% of maximum). Right: Modulation map obtained by continuous rotation of the interferometer around its line-of-sight.

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